the semiring of ideals of \(R\)  under the usual operations, is a residuated lattice denoted \(\mathcal {I}d(R)\) . (In this work, without any assumption, all the rings used, are commutative rings with (*) condition). For example, in 2009, rings \(R\)  for which \(\mathcal {I}d(R)\)  is an MV-algebra known as Lukasiewicz rings were introduced in [1]}. One year later (in 2010), the ring structure for Gödel algebras known as Gödel rings were introduced ([2]}). Recently in 2018, BL-rings were presented in [3]} as rings whose ideals form a BL-algebra. The natural question that one can ask is the following. Are there commutative rings whose lattice of ideals can be equipped with the MTL-algebra's structure without being a BL-algebra?
